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Question

Mathematics Question on Derivatives

lim⁡x→013−21x \displaystyle\lim_{x\rightarrow0} \frac{1}{3-2^{\frac{1}{x}}} is equal to

A

0

B

11

C

12\frac{1}{2}

D

13\frac{1}{3}

Answer

13\frac{1}{3}

Explanation

Solution

lim⁡x→0−13−21/x\displaystyle \lim _{x \rightarrow 0^{-}} \frac{1}{3-2^{1 / x}}
Put x=0−hx =0-h
∴lim⁡h→013−210−h\therefore\displaystyle \lim _{h \rightarrow 0} \frac{1}{3-2^{\frac{1}{0-h}}}
=lim⁡h→013−2−1h=\displaystyle \lim _{h \rightarrow 0} \frac{1}{3-2^{-\frac{1}{h}}}
=13−2−10=13−2−∞=\frac{1}{3-2^{-\frac{1}{0}}}=\frac{1}{3-2^{-\infty}}
=13−0=13=\frac{1}{3-0}=\frac{1}{3}